Extension card · Hesitant
Probabilistic hesitant VIKOR (Li, 2021)
This is the form of VIKOR for situations where a cell carries more than one possible degree, and it is also known how often, or with what probability, each of these degrees is observed. It reduces every cell to a probability-weighted expected value and builds the compromise index over these values.
Base method
VIKOR →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Hesitant →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Three things change. The decision logic stays the same.
Cells. In plain hesitant VIKOR (HF-VIKOR), every cell carries more than one possible value, but how likely each of these values is remains unknown. Here every cell is a set of value-probability pairs: each possible degree comes with its own probability, for example "the value 0.6 with 50 per cent probability, the value 0.8 with 50 per cent probability." The probabilities in a cell cannot sum to more than 1. Weights are given from outside, as single numbers.
Scale equalisation. DecisionMind first reduces every cell, on its own, to an expected value: every degree is multiplied by its own probability and the products summed. After this step, only a single number remains, and HF-VIKOR's problem of equalising set lengths never arises here. The best and worst expected value for every criterion (largest for a benefit, smallest for a cost) is then determined from these expected values.
Distance, score and combination. Group utility S and individual regret R are built exactly as in classical VIKOR. The absolute difference between every alternative's expected value and the best value is taken, divided by the range between the best and worst values, and weighted. S is the sum of these ratios, R is the largest of them. The difference is that the input to this ratio is no longer a raw measurement but the expected value of a probability distribution.
Result and defuzzification. Defuzzification happens at the very start: every cell descends to its expected value before S and R even enter the calculation. This is the exact opposite of the approach in which Fuzzy VIKOR leaves defuzzification to the very end, or HF-VIKOR carries the set through to the last moment. Beneath Q there is no longer any probability-distribution information; only that distribution's mean remains.
DecisionMind fixes, for this family, the expected-value function (a probability-weighted average) and its application to every cell from the very start, and sets the compromise coefficient v at 0.5. In this extension, DecisionMind genuinely computes the two compromise conditions (acceptable advantage, acceptable stability).
How to Read the Output
A lower Q is better, and it is read together with the outcome of the two conditions. The difference is this: this Q rests only on the probability-weighted average of the degrees in every cell. Two cells can share the same expected value while one comes from a narrow probability distribution and the other from a wide one; Q does not distinguish between the two.
Thus instead of writing:
"Probabilistic hesitant VIKOR shows A2 to be the best alternative"
the report should read:
"By expected values, A2 is the sole compromise solution (Q = 0). This figure only reflects the probability-weighted average; how narrow or wide a distribution A2's degrees come from should be examined separately"
When to Prefer This over the Base Method
This extension is suitable when more than one plausible degree exists for the same criterion-alternative pair, and it is known how often each of these degrees was observed. If the probabilities are unknown, and only which degrees are plausible is known, plain hesitant VIKOR (HF-VIKOR) is enough; inventing probabilities adds no information. If criteria are measured, classical VIKOR should be kept. Classical VIKOR's exit condition still applies here: if no compromise whatsoever is acceptable on one criterion, this extension too limits regret but does not eliminate it.
Mistakes Specific to This Extension
Letting the probabilities in a cell sum to more than 1. In that case the expected value is no longer a weighted average but a figure of unclear meaning.
Assuming the probabilities are equal. Giving every degree an equal probability effectively reduces the cell to the plain average of a hesitant set, and hides the real contribution of the probability information.
Forgetting that defuzzification happens at the very start, and assuming set width carries through into S and R. In Fuzzy VIKOR, uncertainty is carried vertex by vertex inside S and R; not here. The expected value is taken at the very first step, and S and R operate on plain crisp numbers.
Skipping the two conditions and declaring the alternative with the smallest Q the "winner." DecisionMind genuinely computes the conditions in this extension. Ignoring this means the most common mistake in classical VIKOR applies here too.
The governing principle is this:
Probabilistic hesitant VIKOR reduces how often each possible degree is observed to an expected value right from the start. If the probabilities do not come from a genuine observation or record, this extension adds nothing over plain hesitant VIKOR.
Cases
The first case is DecisionMind's validation example: a synthetic table of three alternatives and three criteria that can be followed by hand. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Comparing three alternatives with value-probability pairs
DecisionMind's probabilistic hesitant VIKOR validation table compares three alternatives on three criteria; every cell carries two possible degrees and their probability. The third criterion is a cost (lower is better). Weights: C1 = 0.40, C2 = 0.35, C3 = 0.25.
| Alternative | C1 | C2 | C3 (lower is better) |
|---|---|---|---|
| A1 | {0.6 (50%); 0.8 (50%)} | {0.4 (50%); 0.6 (50%)} | {0.5 (50%); 0.7 (50%)} |
| A2 | {0.7 (50%); 0.9 (50%)} | {0.5 (50%); 0.7 (50%)} | {0.3 (50%); 0.5 (50%)} |
| A3 | {0.5 (50%); 0.7 (50%)} | {0.6 (50%); 0.8 (50%)} | {0.4 (50%); 0.6 (50%)} |
The method reduces every cell to its expected value: A1 (0.70; 0.50; 0.60), A2 (0.80; 0.60; 0.40), A3 (0.60; 0.70; 0.50). It weights and sums the normalised differences from each criterion's best and worst expected value (S), and isolates the largest of these (R). It then computes Q with v = 0.5.
| Alternative | S | R | Q |
|---|---|---|---|
| A2 | 0.175 | 0.175 | 0.000 |
| A3 | 0.525 | 0.400 | 0.780 |
| A1 | 0.800 | 0.350 | 0.889 |
The result reads as follows. A2 takes the lowest value on both S and R. Under the acceptable-advantage condition, the Q gap between A2 and A3 is 0.780, above the threshold of 0.5 set for three alternatives. Under the acceptable-stability condition, A2 ranks first on both S and R. Both conditions hold, so the sole compromise solution is A2.
The team's hesitation: what happens if the probability of A1's second degree on the first criterion is raised from 50 per cent to 80 per cent, leaving the remaining 20 per cent on the first degree? A1's expected value rises to 0.76. But A2 still remains first, because A2's expected values are already better on all three criteria. A2's first place is robust to a probability shift of this size.
In the report: "By expected values, A2 is the sole compromise solution (Q = 0); both conditions are satisfied. This result does not change even when A1's probability distribution on the first criterion is redistributed, because A2 holds a higher expected value on all three criteria."
Source: DecisionMind's probabilistic hesitant VIKOR validation example; a fixed table, independent of any sector, built to test the engine's expected-value and S/R/Q arithmetic. The S, R and Q values and the probability-sensitivity check were recomputed independently in Python during this card's preparation.
2. Examination centre: Choosing a new exam-assessment software package
An examination centre will choose among three software packages: Software A, B and C. The criteria are scoring consistency, logistical fit, and cost (lower is better). The centre has tested scoring consistency across different pilot exam sessions. More than one consistency degree was observed for each software package, and how many sessions showed each degree is on record; this can therefore be written as a probabilistic set. The centre has given scoring consistency the highest weight.
The method reduces every cell to its expected value, reverses direction on the cost criterion, sums the weighted differences and isolates the largest of them. With these weights (consistency 0.45, logistics 0.30, cost 0.25), Software A emerges as the sole compromise solution; both conditions hold.
The centre's hesitation: if the weight is concentrated further on consistency (consistency 0.70, logistics 0.15, cost 0.15), Software B moves ahead, because B's expected value on consistency is the highest. At this weight, however, the Q gap between B and A falls below the threshold and the acceptable-advantage condition is not satisfied; the compromise is then a set consisting of B and A together. Once the weight is raised to 0.90, the gap exceeds the threshold and B alone becomes the solution.
In the report: "With the current weights, Software A is the sole compromise solution. Once the weight on consistency is raised to 0.70, Software B moves ahead, but because the acceptable-advantage condition is not satisfied, the compromise is a set consisting of B and A together. Once the weight is raised to 0.90, B alone becomes the solution."
3. What Not to Do
The first error is reading A2's Q = 0 value in Case 1 as "a flawless alternative"; this only expresses the best position among these three alternatives. The second error is entering the probabilities of the two degrees in a cell (50%, 50%) so that they sum to more than 1, for instance as (60%, 60%); the expected value then becomes meaningless. The third error, in Case 2, is declaring Software B, which simply has the smallest Q once the consistency weight is raised to 0.70, the "winner"; this conceals that the acceptable-advantage condition is not satisfied, and the correct outcome is a compromise set.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/phf-vikor
Zhang, S., Xu, Z., & He, Y. (2017). Operations and integrations of probabilistic hesitant fuzzy information in decision making. Information Fusion, 38, 1–11. DOI: 10.1016/j.inffus.2017.02.001
Li, Z. (2021). VIKOR method for evaluating the management quality of teacher education with probabilistic hesitant fuzzy information. Discrete Dynamics in Nature and Society, 2021, Article 1326202. DOI: 10.1155/2021/1326202
Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems (Višekriterijumska optimizacija sistema u građevinarstvu). PhD thesis, University of Belgrade, Faculty of Civil Engineering. (no DOI)
Opricovic, S., & Tzeng, G.-H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2), 445–455. DOI: 10.1016/S0377-2217(03)00020-1
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418